Prefix Sum Array in DSA Python - Build from Scratch

Prefix Sum Array

📖 Scenario: You are working with daily sales data for a small shop. You want to quickly find the total sales from the start of the month up to any given day.

🎯 Goal: Build a prefix sum array that stores cumulative sales totals for each day. This will help you answer total sales queries efficiently.

📋 What You'll Learn

Create a list of daily sales numbers

Create a prefix sum list to store cumulative totals

Use a loop to fill the prefix sum list

Print the prefix sum list

💡 Why This Matters

🌍 Real World

Prefix sum arrays are used in finance, data analysis, and gaming to quickly calculate totals over ranges without repeated addition.

💼 Career

Understanding prefix sums helps in optimizing algorithms and is a common interview topic for software engineering roles.

1

Create the daily sales list

Create a list called daily_sales with these exact values: [10, 20, 15, 30, 25]

DSA Python

# Create the daily_sales list with the given values
# Your code here

Hint

Use square brackets and commas to create the list.

2

Create the prefix sum list

Create a list called prefix_sum with the same length as daily_sales, filled with zeros.

DSA Python

daily_sales = [10, 20, 15, 30, 25]
# Create prefix_sum list with zeros, same length as daily_sales
# Your code here

Hint

Use len() to get the length and multiply a list with zero.

3

Fill the prefix sum list with cumulative totals

Use a for loop with variable i from 0 to len(daily_sales) to fill prefix_sum so that each element is the sum of all previous sales including the current day. For i == 0, set prefix_sum[0] to daily_sales[0]. For others, set prefix_sum[i] to prefix_sum[i-1] + daily_sales[i].

DSA Python

daily_sales = [10, 20, 15, 30, 25]
prefix_sum = [0] * len(daily_sales)
# Fill prefix_sum with cumulative sums using a for loop
# Your code here

Hint

Use range(len(daily_sales)) and an if condition inside the loop.

4

Print the prefix sum list

Print the prefix_sum list to display the cumulative sales totals.

DSA Python

daily_sales = [10, 20, 15, 30, 25]
prefix_sum = [0] * len(daily_sales)
for i in range(len(daily_sales)):
    if i == 0:
        prefix_sum[i] = daily_sales[i]
    else:
        prefix_sum[i] = prefix_sum[i-1] + daily_sales[i]
# Print the prefix_sum list
# Your code here

Hint

Use print(prefix_sum) to show the list.

Practice

1. What does a prefix sum array represent for a given array arr?

easy

A. An array where each element is the product of all elements from the start up to that index in arr.

B. An array where each element is the maximum value found in arr up to that index.

C. An array where each element is the difference between consecutive elements in arr.

D. An array where each element is the sum of all elements from the start up to that index in arr.

Solution

1. Step 1: Understand prefix sum definition

   A prefix sum array stores sums of elements from the start of the original array up to each index.

2. Step 2: Compare options with definition

   Only An array where each element is the sum of all elements from the start up to that index in arr. correctly describes this sum accumulation; others describe different operations.

3. Final Answer:

   An array where each element is the sum of all elements from the start up to that index in arr. -> Option D

4. Quick Check:

   Prefix sum = cumulative sum [OK]

Hint: Prefix sum means adding elements from start to current index [OK]

Common Mistakes:

• Confusing prefix sum with product or max operations
• Thinking prefix sum stores differences
• Assuming prefix sum stores original array values

2. Which of the following Python code snippets correctly initializes a prefix sum array prefix for an input list arr?

easy

A. prefix = [arr[0]]
for i in range(1, len(arr)):
prefix.append(arr[i] - prefix[i-1])

B. prefix = []
for i in range(len(arr)):
prefix.append(arr[i] + arr[i-1])

C. prefix = [arr[0]]
for i in range(1, len(arr)):
prefix.append(prefix[i-1] + arr[i])

D. prefix = [0] * len(arr)
for i in range(len(arr)):
prefix[i] = arr[i] * prefix[i-1]

Solution

1. Step 1: Analyze code for prefix sum logic

   prefix = [arr[0]] for i in range(1, len(arr)): prefix.append(prefix[i-1] + arr[i]) starts prefix with first element and adds current element to previous prefix sum correctly.

2. Step 2: Check other options for errors

   prefix = [] for i in range(len(arr)): prefix.append(arr[i] + arr[i-1]) adds current and previous array elements incorrectly; C multiplies instead of sums; A subtracts instead of sums.

3. Final Answer:

   prefix = [arr[0]]\nfor i in range(1, len(arr)):\n prefix.append(prefix[i-1] + arr[i]) -> Option C

4. Quick Check:

   Prefix sum uses cumulative addition [OK]

Hint: Start prefix with first element, then add previous prefix sum [OK]

Common Mistakes:

• Using multiplication instead of addition
• Adding wrong elements or indices
• Initializing prefix array incorrectly

3. What is the output of the following code?

arr = [3, 1, 4, 1, 5]
prefix = [arr[0]]
for i in range(1, len(arr)):
    prefix.append(prefix[i-1] + arr[i])
print(prefix)

medium

A. [3, 4, 8, 9, 14]

B. [3, 1, 4, 1, 5]

C. [3, 3, 7, 8, 13]

D. [0, 3, 4, 8, 9]

Solution

1. Step 1: Calculate prefix sums step-by-step

   Start with prefix = [3]. Then add 1: 3+1=4, add 4: 4+4=8, add 1: 8+1=9, add 5: 9+5=14.

2. Step 2: Form the final prefix array

   prefix = [3, 4, 8, 9, 14]

3. Final Answer:

   [3, 4, 8, 9, 14] -> Option A

4. Quick Check:

   Sum accumulates correctly [OK]

Hint: Add current element to last prefix sum [OK]

Common Mistakes:

• Confusing prefix array with original array
• Off-by-one errors in indexing
• Incorrect initial prefix value

4. Find the error in this prefix sum code and choose the corrected version:

arr = [2, 4, 6]
prefix = [0]
for i in range(len(arr)):
    prefix.append(prefix[i] + arr[i])
print(prefix)

medium

A. Change prefix = [0] to prefix = [arr[0]] and loop from 1 to len(arr)-1.

B. Change prefix.append(prefix[i] + arr[i]) to prefix.append(prefix[i-1] + arr[i]).

C. Remove the initial zero from prefix and start loop from 0 as is.

D. No error; code works correctly and prints prefix sums.

Solution

1. Step 1: Identify initialization and loop issues

   Initializing prefix with [0] and looping from 0 adds an extra leading zero. Tracing: i=0 append(0+2)=2 -> [0,2]; i=1 append(2+4)=6 -> [0,2,6]; i=2 append(6+6)=12 -> [0,2,6,12]. This produces prefix sums of length n+1 with prefix[0]=0, whereas standard is length n starting from arr[0].

2. Step 2: Correct initialization and loop range

   Start prefix with first element arr[0], then loop from index 1 to end to add sums correctly.

3. Final Answer:

   Change prefix = [0] to prefix = [arr[0]] and loop from 1 to len(arr)-1. -> Option A

4. Quick Check:

   Proper start and loop fix prefix sums [OK]

Hint: Initialize prefix with first element, loop from second element [OK]

Common Mistakes:

• Starting prefix with zero instead of first element
• Looping from zero causing off-by-one errors
• Using wrong prefix index inside loop

5. Given an array arr of length n, how can you use a prefix sum array to find the sum of elements between indices l and r (inclusive) efficiently?

hard

A. Calculate sum as prefix[r] * prefix[l].

B. Calculate sum as prefix[r] - prefix[l-1] if l > 0, else prefix[r].

C. Calculate sum as prefix[r] + prefix[l] always.

D. Calculate sum by looping from l to r and adding elements.

Solution

1. Step 1: Understand prefix sum usage for range sums

   Prefix sum at index r stores sum from start to r. To get sum from l to r, subtract sum from start to l-1.

2. Step 2: Handle edge case when l is zero

   If l is zero, sum is prefix[r] directly since no subtraction needed.

3. Final Answer:

   Calculate sum as prefix[r] - prefix[l-1] if l > 0, else prefix[r]. -> Option B

4. Quick Check:

   Range sum = prefix[r] - prefix[l-1] [OK]

Hint: Subtract prefix before l from prefix at r for range sum [OK]

Common Mistakes:

• Adding prefix sums instead of subtracting
• Ignoring edge case when l = 0
• Using multiplication or looping unnecessarily